Error convergence of quantum linear system solvers

The Harrow-Hassidim-Lloyd (HHL) algorithm remains a paradigmatic quantum routine for solving linear systems and a widely used benchmark in theoretical and experimental studies. Several circuit-level variants have been proposed to simplify its implementation, including constructions in which the clock register is factorized via Hadamard gates to reduce circuit depth and entanglement. In this work, we show that this commonly deployed modification can fundamentally alter the asymptotic behavior of the algorithm and may fail to converge in the limit of large clock size due to persistent phase interference effects. This phenomenon is not a finite-size artifact and compromises the correctness guarantees of the factorized-clock implementation. We provide an analytic characterization of the underlying mechanism and demonstrate that a minimal and experimentally feasible modification suffices to restore convergence while preserving the practical advantages of the simplified circuit. Our results clarify an overlooked correctness issue in a widely used HHL implementation pattern and delineate the conditions under which simplified realizations faithfully reproduce the intended algorithmic behavior.